I'm guessing you've already bought your air tickets, but If you haven't I would suggest arranging things so that you can go to the Amalfi coast first, for reasons of weather. I would probably do Amalfi>Rome>Venice>Florence. However, weather is always a roll of the dice, so if you already have air tickets that would cost a fortune to change, then your order of cities is fine. Amalfi tends to have nice weather for most of October.
You can't "travel by train to the Amalfi". The furthest you can get by train is Sorrento, and then you must take a bus from there. I don't think you can successfully visit both Capri & Positano in the same day, but if you want to try, put Capri in the morning and see Positano later. SInce you will need to pass through Positano leaving the Amalfi to the get to Rome, you might think about staying in Postiano instead of Amalfi. However, you need to first check if ferries from Positano to Capri are still running that late in October. If so, stay in Positano, take the ferry to Capri, return to Amalfi for a quick look around -- and be sure you are on that last bus back to Positano or else you will be stuck with a big taxi bill.
It's between you and God as to whether you want to skip Mass at the Vatican, but you are right that you need to be out of the Amalfi coast the night before. However, if you go sleep in Naples and are on a train by 8am, you should be able to make the 10am Mass -- unless they tell you need to be there hours earlier for security.
It is often best, when traveling as quickly as you intend to travel, to get out of the mentality of traveling on morning trains. What I mean is, consider checking out of a hotel, leaving your bags with the hotel, but doing some more sightseeing in that town before you leave. By the time you decide to go to your next destination, you'll be able to check into your hotel room right away. Many sights and museums in Italy stay open until 7.30pm. So it is possible to, say, stay in Venice in the morning, buy some things for a picnic lunch, get on the train around 1pm, eat your lunch en route to Florence, get to the hotel, shower, and be at a museum or visiting an important church by 5.30. That sort of thing.
PS: Here's one more thought:
When you head to the Amalfi area, get off the train in Naples, store all your luggage (except a toothbrush) in the Naples train station, and take a taxi to the docks to catch a ferry to Capri. Spend the night, next day take a ferry to Amalfi, see the cathedral, then make your way back by bus along the Amalfi drive to Sorrento, get on the train to Naples, pick up your luggage, get on a train to Rome. If it is raining, cancel your visit to the Amalfi coast and spend the time in Naples. Great town and you can book last minute in October no problem. Ditto if you'd rather just go to Rome early.
Keep in mind that the sun sets fairly early at that time of year -- which is another reason I'd want the Amalfi coast earlier rather than later in the trip.